The first question is easy to answer. You can obtain a TIN online by following this link: http://www.irs.gov/businesses/small/article/0,,id=102767,00.html.
The other questions can be answered after consultation with an attorney who can assist you in administrating the estate and after after careful review of the trust. Generally, however, the trustee will be responsible for seeing to it that all required notices are given, the settlor’s debts are paid, all taxes are paid and that all distributions are made in a timely manner to the persons entitled to them. The trustee may incur personal liability for failure to administer a trust in the manner required by law.
How large is the trust estate? If the gross estate (meaning the fair market value of the assets, not reduced by debt) is $2,000,000 or more and the trustor died in 2008, then an IRS Form 706 (Federal Estate Tax Return) is due 9 months after the date of death. That means you must hire both a competent (i) CPA and (ii) probate attorney. (The fact that the attorneys in this field are sometimes referred to as "probate" attorneys does not mean that this estate must go through probate. They are sometimes also referred to as "estate lawyers" or "estate tax lawyers.") The CPA is needed to prepare the decedent's final income tax return, and the income tax return(s) for the trust; the probate attorney is needed to file a copy of the Will with the county, to provide a notice under Probate Code Section 16061.7 to the beneficiaries (as specially defined), to advise you on your responsibilities and to either prepare the 706 or help the CPA do so.
The CPA should file for the taxpayer identification number. The probate attorney will advise you on your responsibilities under (i) California law and (ii) the trust instrument. You should start by reading the trust, in its entirety. A well-drafted trust will describe your recordkeeping responsibilities and the terms under which you are to hold the assets for the minor beneficiaries.
In conclusion, there is material available all over the internet for you to read about your responsibilities. However, there is no substitute for immediately hiring competent counsel. Ask your friends, business colleagues or - best of all - your CPA for a referral to someone with whom they have had good experience in the field of estate planning, probate and/or estate administration.
I don't think I can add any additional value here in terms of what it is that you need to do, but what I do want to emphasize is how complicated this can be, and how convoluted the process can become. Consider that personal liability attaches to any wrongdoing of a Trustee who fails to act properly, and also consider there is an entire area of the law dedicated solely to trust administration.
What I am doing with a client right now is merely consulting him in his duties as trustee. It saves a whole lot more money to hire an attorney as a consultant to teach you and assist you, rather than hire one to do all of the administration tasks. For this simple estate that my client is working on, he will probably only put a few hundred dollars into legal consulting fees (but he already had some experience in this, so that needs to be taken into consideration).
Best of Luck!